At first thought I would think there is. Look at the the dice and calculate the expected value for each open scoring opportunity given n more rolls with n = to 1 or 2 depending what roll you’re on. Always go for the highest expectation.
But now I’m not so sure. In the upper section you need to average 3 dice of each rank to get the 35 point bonus. Many people will purposely go over the average to have a buffer. For example if early on you get a 5 5 5 5 2 do you take a 20 in 5’s or a 22 in quads? Most would say take the 20 since it is two less but opens up more opportunities to get the bonus. Or let’s say you have 6 6 6 1 1. Do you take the three 6’s in the upper zone or the full house . . . or maybe trips? And let’s say you used your Chance square and need to take a 0. Where do you put it?
It appears then that an optimum strategy is not only what dice I keep and which I roll but also how to score them once to roll three times. So does Yahtzee have a theoretical optimum strategy for each roll and each score?
Not a Yahtzee expert, but considering the game, some basic probability knowledge seems sufficient.
It seems like there are 2 different questions, and I’m not sure which is being asked.
Is there a strategy to maximize the expected value of your score? Sure, there has to be one. It’s not hard to see such a strategy must be possible. At each stage, there are a finite number of possible actions, each with an expected value for possible score. You can just pick the action that has the highest expected value. There’s always going to be one.
The existence of such a strategy is thus established but that doesn’t really give clues as what that strategy would be (this is common in math - proving something is possible but not being able to prove how to get there). At worst, you write a program to figure out all possible outcomes, given current roll, score, etc and find the option that produces the highest score, on average. And this is where actual Yahtzee experts might help - maybe the problem has already been studied and such strategies already discovered.
But that leads to the other possible interpretation. Yahtzee is a competitive game. The ‘goal’ is not to maximize your own score but to score higher than your opponent(s). As such, maximizing your ‘return’ for each roll so to speak won’t always be correct.
The ‘optimal’ strategy becomes the strategy that maximizes the probability your score will be higher than your opponents. So, if you are ahead by a fair amount, maybe you want to take a strategy with a slightly lower average score but with much less variance in possible outcomes to give up some points in favor of staying ahead. Or, if you are behind, a strategy that has a lower average score but much higher variance in order to maximize your chances of catching up.
Sure, but it’s orders of magnitude more complicated than maximizing the expected score of each turn. You want to maximize the score of your game.
Your evaluation of your game, after each turn, consists not only of how many points you have, but which boxes you’ve filled up. Not filling up Chance too early is good. Leaving Aces blank for as long as you can (to save it as a free roll if you can clinch the bonus elsewhere) is good. Being over par toward bonus is good. And so on.
It’s a finite game, with a maximum of 195 dice rolls (each of which of course has 6 possible outcomes) and 38 decision points. The 26 decisions after each of the first two rolls branch 32 ways (keep or put back each of 5 dice) and the 12 scoring decisions (which box to fill, after the third roll) branch a diminishing number of ways from 13 through 2 as the game progresses.
That’s a lot of branches, but according to my half-assed googling, Yahtzee is “fully solved”.
Yahtzee is solvable and I’m sure someone has done it, but a full explanation of Yahtzee strategy is too complex to type out. It would be immensely complex to write, because the correct move or roll decision is based on what just happened. In general, though, be sure to fill the “number” scores with good scores (top of the page) early in an effort to get them up to 63 points to get the bonus 35.
The secret to a simpler Yahtzee strategy is to play with enough people that the dominant strategy is to get multiple Yahtzees for the bonus points, although you still have to figure out where to take your points/zeroes for all the rolls that don’t Yahtzee.
work hard on the upper section. Four 5’s or 6’s go in the upper section.
full house seems easy to get, so I usually take the 3 of a kind in the upper section
leave the ones and twos for complete misses later on. For example, if you get a first roll has two one’s and nothing else - reroll all five dice, don’t use up the 1’s.
Getting 4 5s or 4 6s is always a goal of mine to get a bonus in the upper section. You want to be able to absorb getting only 2 of the 3s or 4s and still get the bonus. Keep chance alive as long as you can. Oh, and try to get 5 of a kind.
The upper section takes priority. Those 35 bonus points are like a free roll. Since you probably won’t be getting a Yahtzee, the bonus points will help get the win. So, any early 4-of-a-kinds should go to the upper section. Same with a full house.
Take the full house over a high scoring 3-of-a-kind. A full house and anything later in the 3-of-a-kind will almost always be worth more than any 3-of-a-kind and a 0 full house.
Don’t necessarily save your Chance for later in the game. At the beginning, go for the 5’s and 6’s in the upper section and fill in the Chance if you miss. You’ll probably have a better score in Chance than if you waited to when you need to fill in the 1’s.
Yahtzee is not solvable in the sense used by game theorists. That is, it is not possible to predict the outcome with certainty because the outcome depends on chance (i.e., random dice rolls). The best you can do, as you note, is to assign your rolls as best you can given the current state of the game (i.e., what you and all the other players have rolled so far and how these rolls have been assigned) as well as the expected distribution of future rolls and their fitness for your remaining slots.
(To be extra clear, I’m not saying that your assessment is wrong, only that the terminology you’ve used diverges from that used by people who actually study these things for a living.)
A number of mathematicians and computer scientists have tried to characterize the optimal strategy, and some of them have even written programs that implement these strategies. You can check out, for example:
Games with dice (or other randomizers) can be solved. As you note, you can “assign your rolls as best you can given the current state of the game.” That is called “solution.”
Do you have a cite for your claim? (Other than one Wikipedia article which does seem to agree with you.)
I guess I don’t know anyone who “studies these things for a living” (unless preparing theses for graduate degrees is considered “for a living”) but, were gambling still allowed at SDMB, I might bet money that “professional game solvers” regard the solution of a game like Pai Gow or Blackjack as a “solution.”
I wouldn’t call that a solution, but rather a strategy for maximizing the chances of winning.
Did you check out the references in the article? I suspect that most of them to game theory articles will be using the term in this way. The thesis by Allis certainly does—he distinguishes between three types of solvability, from ultra-weak (“For the initial position(s), the game-theoretic value has been determined”) to strong (“For all legal positions, a strategy has been determined to obtain the game-theoretic value of the position, for both players, under reasonable resources.”). Note here that “game-theoretic value” is not some number representing the probability of winning, but rather a binary “win” or “lose” that is guaranteed through perfect play.
Maybe you are right that game players may use the term differently; I was speaking only about game theorists. (And sure, there are people who do this for a living, in their capacity as researchers, professors, or economists. There are entire scholarly journals devoted to the field, such as the International Journal of Game Theory). But for what it’s worth, at least some game designers use the strict mathematical definition. For example, Brenda Brathwaite and Ian Schreiber, writing in their book Challenges for Game Designers, write that “a game is ‘solvable’ if the entire possibility space is known ahead of time and can be exploited such that a specific player, playing correctly, can always win (or draw)” (emphasis mine). By that definition, many games of chance, including Yahtzee, are not solvable.
I’m pretty sure Yahtzee is, if not solved, basically solved. See, for instance Exact Algorithms for Solving Stochastic Games, Yahtzee is not too far off from a modification of the game described in the introduction on a conceptual level. Player 1 and player 2 make a move simultaneously each turn, causing them to exchange a certain number of points, and then with some probability the game changes to another state. You can modify the scoring system of Yahtzee to be about exchanging points from a pool of total points without really changing the game, and the “random space” you land on each turn can easily be described with dice configurations.
The main difference is that player 2 has a chance to observe player 1 before making their move, and minor changes can have a major effect on game theory, but IMO in this case the information transfer is not quite important enough to change it from a solvable to an unsolvable class of problems. In fact, you could probably make the problems nearly isomorphic by saying that the player’s action space is limited by the current “space” they’re in, and that the transition function from all even number turns can only land you on “spaces” where player 2 can only do no-op, and vice versa for odd turns.
An Optimal Strategy for Yahtzee. It’s for solitaire Yahtzee, but tbh “competitive” Yahtzee is just two people playing solitaire Yahtzee and then comparing points at the end, since players can’t interfere with each others dice or score sheets at all.
It’s not quite true that competitive Yahtzee is the same as solitaire Yahtzee. There can be situations, particularly near the end of the game, where one player is so far ahead that the only way the other player(s) have a chance of catching up is to take risks—risks that they would never take when trying to maximize their expected score in a solitaire game.
For example, consider a two-player game that is two turns away from the end. Player 1 is 30 points behind Player 2, and has only “Full house” (25 points) and “Yahtzee” (50 points) remaining on their score card. If they roll something that is neither a full house nor a yahtzee, then the only hope they have of winning the game is to write a 0 in “Full house” and to roll a Yahtzee on their next turn. By contrast, if they were playing solitaire, the correct strategy would be to write a 0 in “Yahtzee” since there is a much higher probability of them rolling a full house on their next turn and thus finishing the game with a higher total score.